题目

Problem

The diagram shows part of the curve with equation $y=\frac{1}{2}x+\frac{4}{x}$ and the line $y=4.5$ .

Find the exact volume of the solid formed when the shaded region is rotated through $360^\circ$ about the $x$ -axis.

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题目中文翻译

图中显示的是方程为 $y=\frac{1}{2}x+\frac{4}{x}$ 的曲线的一部分，以及直线 $y=4.5$ 。

求阴影区域绕 $x$ 轴旋转 $360^\circ$ 后形成的立体的精确体积。

解答

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先求曲线与直线的两个交点。阴影绕 $x$ 轴旋转后，每个截面是一个圆环，外半径恒为 $4.5$ ，内半径是曲线的高度。

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At an intersection,

\frac x2+\frac4x=\frac92 \quad\Longrightarrow\quad (x-1)(x-8)=0.

Between $1$ and $8$ the curve lies below the line and above the axis. The volume is

\begin{align*} V=&\,\pi\int_1^8\Big[\frac{81}{4}-\Big(\frac x2+\frac4x\Big)^2\Big]\,\mathrm{d}x\\ =&\,\pi\int_1^8\Big(\frac{65}{4}-\frac{x^2}{4}-\frac{16}{x^2}\Big)\,\mathrm{d}x\\ =&\,\pi\Big[\frac{65x}{4}-\frac{x^3}{12}+\frac{16}{x}\Big]_1^8\\ =&\,\pi\Big(\frac{268}3-\frac{193}6\Big) =\frac{343\pi}6. \end{align*}